
where \(\alpha _{u(\cdot , v)}^{\text{pre}}\)  needs to be approximated for the specific types \(v\) .
We show next that the mapping on \(\alpha \) -Rank distributions given by \(f(\alpha , u(\cdot , v))\)  is continuously differentiable in both arguments if the utilities are continuously differentiable in the types. For our purposes in modeling uncertainty in vNM functions, it means that the solution is not too sensitive to changes in the vNM functions, which is a desirable property because for a large set \(\mathcal {V}\)  one relies on approximation methods. The approximation technique's error bounds often rely on the integrand's variance, which is easier to bound for continuously differentiable functions. For example, variance reduction methods, such as quasi-Monte-Carlo integration, only have a meaningful bound for continuous integrands [1]}.
Even though the limiting \(\alpha \) -Rank distribution \(g(u(\cdot , v))\)  is not necessarily even continuous, each evaluation of \(f(\alpha , u(\cdot , v))\)  gives an indication for a small neighborhood. The experiments in Section REF  show a smooth transition of mass among the strategy profiles with changing utilities.
